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Motivated by the reservoir engineering concept of the productivity index of a producing oil well in an isolated reservoir, we analyze a time dependent functional, diffusive capacity, on the solutions to initial boundary value problems for a parabolic equation. Sufficient conditions providing for time independent diffusive capacity are given for different boundary conditions. The dependence of the constant diffusive capacity on the type of the boundary condition (Dirichlet, Neumann or third-type boundary condition) is investigated using a known variational principle and confirmed numerically for various geometrical settings. An important comparison between two principal constant values of a diffusive capacity is made, leading to the establishment of criteria when the so-called pseudo-steady-state and boundary-dominated productivity indices of a well significantly differ from each other. The third type boundary condition is shown to model the thin skin effect for the constant wellbore pressure production regime for a damaged well. The questions of stabilization and uniqueness of the time independent values of the diffusive capacity are addressed. The derived formulas are used in numerical study of evaluating the productivity index of a well in a general three-dimensional reservoir for a variety of well configurations.

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The use of foam-forming surfactants offers promise to improve sweep efficiency and mobility control for enhanced oil recovery (EOR). This thesis provides an in depth understanding of transport of surfactant and foam through porous media using a combination of laboratory experiments and numerical simulations. In particular, there are several issues in foam EOR processes that are examined. These include screening of surfactant adsorption onto representative rock surfaces, modeling of foam flow through porous media, and studying the effects of surface wettability and porous media heterogeneity. Surfactant adsorption onto rock surfaces is a main cause of foam chromatographic retardation as well as increased process cost. Successful foam application requires low surfactant adsorption on reservoir rock. The focus of this thesis is natural carbonate rock surfaces, such as dolomite. Surfactant adsorption was found to be highly dependent on electrostatic interactions between surfactants and rock surface. For example, the nonionic surfactant Tergitol 15-S-30 exhibits low adsorption on dolomite under alkaline conditions. In contrast, high adsorption of cationic surfactants was observed on some natural carbonate surfaces. XPS analysis reveals silicon and aluminum impurities exist in natural carbonates, but not in synthetic calcite. The high adsorption is due to the strong electrostatic interactions between the cationic surfactants and negative binding sites in silica and/or clay. There are a number of commercial foam simulators, but an approach to estimate foam modeling parameters from laboratory experiments is needed to simulate foam transport. A one-dimensional foam simulator is developed to simulate foam flow. Chromatographic retardation of surfactants caused by adsorption and by partition between phases is investigated. The parameters in the foam model are estimated with an approach utilizing both steady-state and transient experiments. By superimposing contour plots of the transition foam quality and the foam apparent viscosity, one can estimate the reference mobility reduction factor (fmmob) and the critical water saturation (fmdry) using the STARS foam model. The parameter epdry, which regulates the abruptness of the foam dry-out effect, can be estimated by a transient foam experiment in which 100% gas displaces surfactant solution at 100% water saturation. Micromodel experiments allow for pore-level visualization of foam transport. We have developed model porous media systems using polydimethylsiloxane. We developed a simple method to tune and pattern the wettability of polydimethylsiloxane (PDMS) to generate porous media models with specific structure and wettability. The effect of wettability on flow patterns is observed in gas-liquid flow. The use of foam to divert flow from high permeable to low permeable regions is demonstrated in a heterogeneous porous micromodel. Compared with 100% gas injection, surfactant-stabilized foam effectively improves the sweep of the aqueous fluid in both high and low permeability regions of the micromodel. The best performance of foam on fluid diversion is observed in the lamella-separated foam regime, where the presence of foam can enhance gas saturation in the low permeable region up to 45.1% at the time of gas breakthrough. In conclusion, this thesis provides new findings in surfactant adsorption onto mineral surfaces, in the methodology of estimating foam parameters for reservoir simulation, and in micromodel observations of foam flow through porous media. These findings will be useful to design foam flooding in EOR processes.

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We investigate how the formation of heterogeneity and structures in flows through explicit porous microstructures depends upon the geometric and topological observables of the porous medium. Using direct numerical simulations of single-phase, isothermal, laminar fluid flow through realistic three-dimensional stochastically generated pore structures, hereafter referred to as pore spaces, the characteristics of the resulting steady state velocity fields are related to physical characteristics of the pore spaces. The results suggest that the spatially variable resistance offered by the geometry and topology of the pore space induces a highly heterogeneous fluid velocity field therein. Focus is placed on three different length scales: macroscopic (cm), mesoscopic (mm), and microscopic (microns). At the macroscopic length scale, volume averaging is used to relate porosity, mean hydraulic radius, and their product to the permeability of the pore space. At the mesoscopic scale, the effect of a medium's porosity on fluid particle trajectory attributes, such as passage time and tortuosity, is studied. At the final length scale, that of the microscopic in-pore fluid dynamics, finite time Lyapunov exponents are used to determine expanding, contracting, and hyperbolic regions in the flow field, which are then related to the local structure of the pore space. The results have implications to contaminant transport, mixing, and how chemical reactions are induced at the pore-scale. A description of the adopted numerical methods to simulate flow and generate the pore space are provided as well.

In this dissertation, we present multiscale analytical solutions, in the weak sense, to the generalized Laplace's equation in Ω ⊂ Rⁿ, subject to periodic and nonperiodic boundary conditions. They are called multiscale solutions since they depend on a coefficient which takes a wide possible range of scales. We define forms of nonseparable coefficient functions in Lᵖ(Ω) such that the solutions are valid for the periodic and nonperiodic cases. In the periodic case, one such solution corresponds to the auxiliary cell problem in homogenization theory. Based on the proposed analytical solution, we were able to write explicitly the analytical form for the upscaled equation with an effective coefficient, for linear and nonlinear cases including the one with body forces. This was done by performing the two-scale asymptotic expansion for linear and nonlinear equations in divergence form with periodic coefficient. We proved that the proposed homogenized coefficient satisfies the Voigt-Reiss inequality. By performing numerical experiments and error analyses, we were able to compare the heterogeneous equation and its homogenized approximation in order to define criteria in terms of allowable heterogeneity in the domain to obtain a good approximation. The results presented, in this dissertation, have laid mathematical groundwork to better understand and apply multiscale processes under a deterministic point of view.

Transient multi-phase flow problems in porous media are ubiquitous in engineering and environmental systems and processes; examples include heat exchangers, reservoir simulation, environmental remediation, magma flow in the earth crust and water management in porous electrodes of PEM fuel cells. This thesis focuses on the development of accurate and computationally efficient numerical models to simulate such flows. The research challenges addressed in this work fall in two areas. For a numerical standpoint, conventional numerical methods including Newton-Raphson linearization and a simple upwind scheme do not always provide the required computational efficiency or sufficiently accurate resolution of the flow field. From a modelling perspective, closure schemes required in volume-averaged formulations, such as the generalized Leverett J function for capillary pressure, are specific to certain media (e.g. lithologic media) and are not valid for fibrous porous media, which are of central interest in fuel cells. This thesis presents a set of algorithms that are integrated efficiently to achieve computations that are more than two orders of magnitude faster compared to traditional techniques. The method uses an adaptive operator splitting method based on an a posteriori criterion to separate the flow from the transport equations which eliminates unnecessary and costly solution of the implicit pressure-velocity term at every time step; adaptive meshing to reduce the size of the discretized problem; efficient block preconditioned solver techniques for fast solution of the discrete equations; and a recently developed artificial diffusion strategy to stabilize the numerical solution of the transport equation. The significant improvements in accuracy and efficiency of the approach is demosntrated using numerical experiments in 2D and 3D. The method is also extended to advection-dominated problems to specifically investigate two-phase flow in heterogeneous porous media involving capillary transport. Both hydrophilic and hydrophobic media are considered, and insights relevant to fuel cell electrodes are discussed.

[pt] O presente trabalho tem como objetivo a modelagem de escoamentos através de meios porosos, sob o ponto de vista da Teoria Contínua de misturas. O fluido e o sólido, que compõe o meio poroso, são tratados como constituintes contínuos de uma mistura binária, onde não ocorrem reações químicas. Em todas as situações aqui tratadas o fluido é suposto Newtoniano e incompressível, enquanto o meio poroso é rígido, homogêneo e isotrópico. O trabalho pode ser dividido em duas partes principais. Na primeira são modelados escoamentos através de regiões contendo meios porosos saturados e regiões onde só existe o fluido. São discutidas condições de compatibilidade sobre as interfaces, que separam as regiões, e é estabelecido um modelo para escoamentos, nos quais não exista fluxo de massa através das interfaces. A segunda parte trata de escoamentos em meios porosos insaturados, onde é preciso se considerar o efeito de forças capilares. Nesta parte é estabelecido um modelo e são simuladas situações unidimensionais. São estudados vários casos entre eles o enchimento de uma placa porosa, com e sem efeitos de atrito e de forças gravitacionais. A obtenção de resultados, nestes casos, exige a solução numérica de um sistema hiperbólico não-linear de equações diferenciais. / [en] This work aims to a modelling of flow through a porous media based upon the Continuum Theory of Mixtures. The fluid and the solid, which composes the porous media, are assumed as continuous constituent of a binary mixture where chemical reactions do not occur. In all situations here considered, the fluid is assuned Newtonian and incompressíble, while the porous media is rigid, homogeneus and isotropic. This work can be divided in two main parts. In the first one, flows are modelled through regions containing saturated porous media and regions where there is nothing but the fluid. Conditions of compatibility in the interfaces that divide the regions are discussed and a flow modelling is stablished where there are no crosaflow through the interfaces. The second part is concerned with flows in unsaturated porous media, where the effect of capillery pressure is considered. In this Part a model is stablished and unidimensíonal situations are simulated. Several cases are studied and the filling-up of a porous plate is among them, with and without frictíon effect and gravitational forces. The obtainment of results, in such cases, requires the numeric solution of a non-linear hyperbolíc system of differential equations.